What is the slope of the line tangent to $f(x) = -x^{2}+2x+7$ at $x = 4$ ?
The slope of the tangent line is $ \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}$ $ = \lim_{\Delta x \to 0} \frac{(-(x+\Delta x)^{2}+2(x+\Delta x)+7) - (-x^{2}+2x+7)}{\Delta x}$ $ = \lim_{\Delta x \to 0} \frac{(-(x^{2}+2x \Delta x+\Delta x^{2})+2(x+\Delta x)+7) - (-x^{2}+2x+7)}{\Delta x}$ $ = \lim_{\Delta x \to 0} \frac{-x^{2}-2(x \Delta x)-\Delta x^{2}+2x+2(\Delta x)+7+x^{2}-2x-7}{\Delta x}$ $ = \lim_{\Delta x \to 0} \frac{-2(x \Delta x)-\Delta x^{2}+2(\Delta x)}{\Delta x}$ $ = \lim_{\Delta x \to 0} -2x-\Delta x+2$ $ = -2x+2$ $ = (-2)(4)+2$ $ = -6$